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Slide 1 / 160 Slide 2 / 160 Pre-alc onics 2015-03-24 www.njctl.

to onic Sections ircles Midpoint and istance Formula Hyperbolas Recognizing onic Sections from the General Form Return to Table of

1 Slide 1 / 160 Slide 2 / 160 Pre-alc onics Slide 3 / 160 Slide 4 / 160 Table of ontents click on the topic to go to that section Review of Midpoint and istance Formulas Intro to onic Sections ircles Midpoint and istance Formula Hyperbolas Recognizing onic Sections from the General Form Return to Table of ontents Slide 5 / 160 Slide 6 / 160 Midpoint and istance Formula The Midpoint Formula Give points (x1,y1) and (x2,y2), the point midway between and is Examples: Find the midpoint of the segment with the given endpoints.

2 Slide 7 / 160 Slide 8 / 160 Midpoint and istance Formula Midpoint and istance Formula 1 Find the midpoint of K(1,8) & L(5,2). (2,3) 2 Find the midpoint of H(-4, 8) & L(6, 10). (1,9) (3,5) (2,18) (-2,-3) (-2,-18) (-3,-5) (-1,-9) Slide 9 / 160 Slide 10 / 160 Midpoint and istance Formula 3 Given the midpoint of a segment is (4, 9) and one endpoint is (-3, 10), find the other midpoint. (-10, 8) (11, 8) (-10, 11) (.5, 9.5) Slide 11 / 160 Slide 12 / 160 Midpoint and istance Formula 4 What is the distance between (2, 4) and (-1, 8)?

Midpoint and istance Formula Slide 13 / 160 5 What is the distance between (0, 7) and (5, -5)?

3 Midpoint and istance Formula Slide 13 / What is the distance between (0, 7) and (5, -5)? Midpoint and istance Formula Slide 14 / 160 Note: The distance between points and can be notated as Midpoint and istance Formula Slide 15 / Given ( 4, 5) and (x, 1) and =5, find all of the possible values of x. Slide 16 / E 0 F 1 G 3 H 5 I 7 J 9 Intro to onic Sections Return to Table of ontents Slide 17 / 160 Slide 18 / 160 Intro to onic Sections onic Sections come from cutting through 2 cones, which is called taking cross sections. onic Sections are often times not functions because they do not pass the Vertical Line Test. Intro to onic Sections ircle comes from cutting parallel to the "base". The term base is mis-leading because cones continue on, like lines.

Slide 19 / 160 Slide 20 / 160 Intro to onic Sections Intro to onic Sections n Ellipse comes from cutting skew (diagonal) to the "base".

Slide 21 / 160 Slide 22 / 160 Intro to onic Sections Hyperbola comes from cutting the cones perpendicular to the "bases".

Return to Table of ontents Slide 23 / 160 Slide 24 / 160 s we've studied earlier, come from a quadratic equation of the form y=ax 2 +bx+c and

4 Slide 19 / 160 Slide 20 / 160 Intro to onic Sections Intro to onic Sections n Ellipse comes from cutting skew (diagonal) to the "base". Parabola comes from cutting the cone an intersecting the "base" and parallel to a side. Slide 21 / 160 Slide 22 / 160 Intro to onic Sections Hyperbola comes from cutting the cones perpendicular to the "bases". This is the only cross section that intersects both cones. Return to Table of ontents Slide 23 / 160 Slide 24 / 160 s we've studied earlier, come from a quadratic equation of the form y=ax 2 +bx+c and have a "U" shaped graph. nother helpful form of the equation is called Standard Form. Standard Form is (x - h) 2 = 4p(y - k), where (h,k) is the vertex. This is also called Vertex Form. 7 What is the vertex of (3, 2) (-3, -2) (2, 3) (-2, -3) Example: What is the vertex of: (x - 4) 2 = -3(y - 5) (x + 7) 2 = 2(y - 2) (x -3) 2 = y

5 Slide 25 / 160 Slide 26 / What is the vertex of (3, 2) (-3, -2) (2, 3) (-2, -3) 9 What is the vertex of (3, 2) (-3, -2) (2, -3) (-2, -3) Slide 27 / 160 Slide 28 / 160 Slide 29 / 160 Slide 30 / What is the vertex of (3, 2) (-3, 2) (2, 3) (-2, -3) 11 What is the vertex of (3, 2) (-3, -2) (2, 3) (-2, -3)

6 Slide 31 / 160 Slide 32 / 160 onverting from General Form to Standard Form Note: To convert into Standard Form, we use a process called ompleting the Square. Slide 33 / 160 Steps: 1) Group the quadratic and its linear term on one side, and move the other linear and constant terms to the other side. 2) If there is a number in front of the quadratic, factor it out of the group. 3) Take the number in front of the linear term, divide it in half and square it. 4) dd this number inside the parenthesis; multiply it by the number you factored out in step two, and add it to the other side of the equation as well. 5) Factor the quadratic function inside the parenthesis Slide 34 / 160 Example: Find the vertex of the parabola Slide 35 / 160 Slide 36 / 160

Slide 37 / 160 Slide 38 / 160 Slide 39 / 160 Slide 40 / 160 17 What is the vertex of y 2-10y - x + 29 = 0?

7 Slide 37 / 160 Slide 38 / 160 Slide 39 / 160 Slide 40 / What is the vertex of y 2-10y - x + 29 = 0? (4, 5) 18 What is the vertex of (4, 5) (-4, 5) (-4, 5) (-5, 4) (-5, 4) (5, 4) (5, 4) Slide 41 / 160 Slide 42 / 160 onverting from General Form to Standard Form 19 What should be factored out of (4y 2-8y + ) = x - 9 +? } +18 } -12

Slide 43 / 160 Slide 44 / 160 20 What value completes the square of 4(y 2-2y + ) = x - 9 +? 21 What value should follow "-9" in 4(y 2-2y + ) = x - 9 +?

8 Slide 43 / 160 Slide 44 / What value completes the square of 4(y 2-2y + ) = x - 9 +? 21 What value should follow "-9" in 4(y 2-2y + ) = x - 9 +? Slide 45 / 160 Slide 46 / Which is the correct standard form of 4(y 2-2y + ) = x What should be factored out of (-5x 2-20x + ) = y - 7 +? Slide 47 / 160 Slide 48 / What value completes the square of -5(x 2 + 4x + ) = y - 7 +? 25 What value should follow "-7" in -5(x 2 + 4x + ) = y - 7 +?

Slide 49 / 160 26 Which is the correct standard form of (-5x 2-20x + ) = y - 7 + Slide 50 / 160 Focus and irectrix of a Parabola Every point on the parabola is the same distance from the directrix

9 Slide 49 / Which is the correct standard form of (-5x 2-20x + ) = y Slide 50 / 160 Focus and irectrix of a Parabola Every point on the parabola is the same distance from the directrix and the focus. L 1=L 2 L 1 L 2 Focus xis of Symmetry irectrix The focal distance is the distance from the vertex to the focus, which is the same as the distance from the vertex to the directrix. Slide 51 / 160 Eccentricity of a Parabola L 1=L 2 Slide 52 / 160 Parts of a Parabola Whether a quadratic has the x 2 or y 2, they have the same parts. ax 2 +bx+dy+e=0 cy 2 +dy+bx+e=0 L 1 L 2 Focus Focus Vertex irectrix xis of Symmetry Focus Vertex irectrix irectrix xis of Symmetry Slide 53 / 160 Slide 54 / 160 Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity?

10 Slide 55 / 160 Graph the equation from the last example. Slide 56 / 160 Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity? Slide 57 / 160 Slide 58 / 160 Graph Slide 59 / 160 Slide 60 / 160 Graph 27 Given the following equation, which direction does it open? UP OWN LEFT RIGHT

11 Slide 61 / 160 Slide 62 / Where is the vertex for the following equation? (-3, 4) 29 What is the equation of the axis of symmetry for the following equation? (3, 4) y = 3 (4, 3) y = -3 (4, -3) x = 4 x = -4 Slide 63 / 160 Slide 64 / What is the focal distance in the following equation? 31 What is the equation of the directrix for the following equation? y = 2 y = -4 x = 3 x = -5 Slide 65 / 160 Slide 66 / Where is the focus for the following equation? (-3, 5) (3, 5) (5, 3) (5, -3) 33 What is the eccentricity of the following conic section?

12 Slide 67 / 160 Slide 68 / 160 Slide 69 / 160 Slide 70 / 160 Slide 71 / 160 Slide 72 / 160

13 Slide 73 / 160 Slide 74 / 160 Slide 75 / 160 Slide 76 / Where is the vertex for the following equation? (0, 4) 43 What is the equation of the axis of symmetry for the following equation? (0, -4) y = 0 (4, 0) y = -0 (-4, 0) x = 4 x = -4 Slide 77 / 160 Slide 78 / What is the focal distance in the following equation? 45 What is the equation of the directrix for the following equation? y = 0 y = -4 x = 8 x = 0

14 Slide 79 / 160 Slide 80 / Where is the focus for the following equation? (4, 8) (-4, 4) (4, 4) (4, -4) 47 What is the eccentricity of the following conic section? Slide 81 / 160 Slide 82 / 160 ircles Return to Table of ontents Slide 83 / 160 Slide 84 / 160

ircles Slide 85 / 160 48 Write the equation

and radius 7 ircles Slide 87 / 160 ircles

circle with center (-2, 1) and radius 51

15 ircles Slide 85 / Write the equation of the circle with center (5, 2) and radius 6 ircles Slide 86 / Write the equation of the circle with center (-5, 0) and radius 7 ircles Slide 87 / 160 ircles Slide 88 / Write the equation of the circle with center (-2, 1) and radius 51 What is the center and radius of the following equation? Slide 89 / 160 Slide 90 / 160 ircles 53 What is the center and radius of the following equation?

16 Slide 91 / 160 Slide 92 / 160 ircles ircles 54 What is eccentricity of a circle? Ex: Write the equation of the circle that meets the following criteria: iameter with endpoints (4, 7) and (-2, -1). Since the midpoint of the diameter is the center use the midpoint formula. The radius is distance from the center to either of the given points. Slide 93 / 160 Slide 94 / 160 ircles Ex: Write the equation of the circle that meets the following criteria: enter (1, -2) and passes through (4, 6) Since we know the center we only need to find the radius. The radius is the distance from the center to the point. ircles Ex: Write the equation of the circle that meets the following criteria: enter at (-5, 6) and tangent to the y-axis. "Tangent to the y-axis" means the circle only touches the y-axis at one point. Look at the graph. Slide 95 / 160 Slide 96 / 160 ircles Write the equation of the circle in standard form that meets the following criteria: omplete the square for the x's

17 Slide 97 / 160 Slide 98 / 160 ircles ircles 55 What is the equation of the circle that has a diameter with endpoints (0, 0) and (16, 12)? 56 What is the equation of the circle with center (-3, 5) and contains point (1, 3)? Slide 99 / 160 Slide 100 / 160 ircles 57 What is the equation of the circle with center (7, -3) and tangent to the x-axis? Slide 101 / 160 Slide 102 / 160 Return to Table of ontents

18 Slide 103 / 160 Slide 104 / 160 n ellipse is the set of points the same total distance from 2 points. In this example, In this graph F 1 and F 2 are foci. (Plural of focus) They lie on the major axis. (The longest distance) P The shortest distance is the minor axis. Where the axes intersect is the ellipse's center. P s point moves along the ellipse, L 1 and L 2 will change but their sum will stay ten. The more elongated the ellipse the closer the eccentricity is to 1. The closer an ellipse is to being a circle, the closer the eccentricity is to 0. (0 < e < 1) Slide 105 / What letter or letters corresponds with ellipse's center? E E Slide 106 / What letter or letters corresponds with ellipse's foci? E E Slide 107 / 160 Slide 108 / What letter or letters corresponds with ellipse's major axis? E E 63 Which choice best describes an ellipse's eccentricity? e = 0 0< e < 1 e = 1 e > 1

19 Slide 109 / 160 Slide 110 / 160 Slide 111 / 160 Slide 112 / What is the center of (9, 4) (5, 6) (-5, -6) (3, 2) Slide 113 / 160 Slide 114 / How long is the major axis of 9 66 How long is the minor axis of

20 Slide 115 / 160 Slide 116 / Name one foci of 68 Name one foci of Slide 117 / 160 Slide 118 / 160 Graphing an Ellipse Find and graph the center Find the length and direction of the major and minor axes From the center go half the length the axis from the center for each Graph the ellipse The center is (4, -2) The major axis is 6 units and horizontal The minor axis is 4 units and vertical Slide 119 / 160 Slide 120 / 160 What is equation of an ellipse with foci (3, -2) and (3, 6) and minor axis of length 8?

Slide 121 / 160 69 Given that an ellipse has foci (4, 1) and (-4, 1) and major axis of length 10, what is the center of the ellipse?

21 Slide 121 / Given that an ellipse has foci (4, 1) and (-4, 1) and major axis of length 10, what is the center of the ellipse? Slide 122 / Given that an ellipse has foci (4, 1) and (-4, 1) and major axis of length 10, in which direction is the ellipse elongated? (8, 2) (0, 2) (0, 1) (-8, 1) horizontally vertically obliquely it is not elongated Slide 123 / 160 Slide 124 / Given that an ellipse has foci (4, 1) and (-4, 1) and major axis of length 10, how far is it from the center to an endpoint of the major axis? Given that an ellipse has foci (4, 1) and (-4, 1) and major axis of length 10, which equation would be used to find the distance from the center to an endpoint of the minor axis? Slide 125 / 160 Slide 126 / Given that an ellipse has foci (4, 1) and (-4, 1) and major axis of length 10, find a.

22 Slide 127 / 160 Slide 128 / 160 onverting to Standard Form complete the square for x and/or y factor the x's and y's divide by the constant Ex: Ex: Slide 129 / 160 Slide 130 / onvert the following ellipse to standard form. Slide 131 / 160 Slide 132 / 160 Hyperbolas Return to Table of ontents

Slide 133 / 160 Slide 134 / 160 Hyperbolas The standard form of a horizontal hyperbola is Hyperbolas The standard form of a vertical hyperbola is Hyperbolas Slide 135 / 160 To graph a hyperbola in

diagonals of the rectangle decide if hyperbola goes left & right or up & down left & right: the "x term" is first up & down: the "y term" is first graph hyperbola Hyperbolas Example: Graph The center

23 Slide 133 / 160 Slide 134 / 160 Hyperbolas The standard form of a horizontal hyperbola is Hyperbolas The standard form of a vertical hyperbola is Hyperbolas Slide 135 / 160 To graph a hyperbola in standard form: graph (h,k) as center of graph go a right and left of the center, and b up and down make a rectangle through the four points from previous step draw asymptotes that contain the diagonals of the rectangle decide if hyperbola goes left & right or up & down left & right: the "x term" is first up & down: the "y term" is first graph hyperbola Hyperbolas Example: Graph The center of the rectangle is? From the center move left/right? From the center move up/down? The hyperbola opens? What are the slopes of the asymptotes? How does this relate to a and b? Why? Slide 136 / 160 Hyperbolas Slide 137 / 160 Slide 138 / 160 Example: Graph The center of the rectangle is? From the center move left/right? From the center move up/down? The hyperbola opens?

24 Slide 139 / 160 Slide 140 / 160 Slide 141 / 160 Slide 142 / 160 Slide 143 / 160 Slide 144 / 160

Slide 145 / 160 Slide 146 / 160 Hyperbolas Standard Form of an Hyperbola The Foci are equidistant from the center in the horizontal direction if the x-term comes first, or in the vertical direction

25 Slide 145 / 160 Slide 146 / 160 Hyperbolas Standard Form of an Hyperbola The Foci are equidistant from the center in the horizontal direction if the x-term comes first, or in the vertical direction if the y-term comes first. The distance from the center to the foci is In this example, the focal distance is nd their location is at and Slide 147 / 160 Slide 148 / 160 Hyperbolas 85 What is the focal distance for the following equation? Hyperbolas 86 What is the location of one of the foci for this hyperbola? (-13,-6) (10,-6) (-10,-6) (13,-6) Hyperbolas Slide 149 / 160 Slide 150 / What is the location of one of the foci for this hyperbola? (3,7) (3,19) (3,-7) (3,13)

26 Slide 151 / 160 Slide 152 / 160 Recognizing onic Sections from the General Form Return to Table of ontents Slide 153 / 160 Slide 154 / 160 Slide 155 / 160 Slide 156 / 160

27 Slide 157 / 160 Slide 158 / 160 Slide 159 / 160 Slide 160 / 160

Pre-Calc. Slide 1 / 160. Slide 2 / 160. Slide 3 / 160. Conics Table of Contents. Review of Midpoint and Distance Formulas

Pre-Calc. Slide 1 / 160. Slide 2 / 160. Slide 3 / 160. Conics Table of Contents. Review of Midpoint and Distance Formulas Slide 1 / 160 Pre-Calc Slide 2 / 160 Conics 2015-03-24 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 160 Review of Midpoint and Distance Formulas Intro to Conic Sections